Obtuse Triangle
GeometryAn obtuse triangle has one interior angle measuring greater than 90 degrees.
Formula
a^2 + b^2 < c^2 \text{ (where } c \text{ is the longest side)}
Definition
An obtuse triangle has one angle larger than $90^\circ$. Only one angle can be obtuse in a triangle (there is not enough room for two).
Example
A triangle with angles $120^\circ$, $30^\circ$, and $30^\circ$ is obtuse because $120 > 90$. If you draw a very flat, wide triangle, the wide angle at the top will likely be obtuse.
Key Insight
A triangle can have at most one obtuse angle because the three angles must add to $180^\circ$. If two angles were each greater than $90^\circ$, their sum alone would exceed $180$, which is impossible.
Definition
An obtuse triangle has exactly one angle greater than $90^\circ$. For sides $a, b, c$ ($c$ being the longest): the triangle is obtuse iff $c^2 > a^2 + b^2$. The circumcenter and orthocenter of an obtuse triangle both lie outside the triangle.
Example
Triangle with sides $5, 5, 9$: check $c^2$ vs $a^2 + b^2$: $81$ vs $25 + 25 = 50$. Since $81 > 50$, the triangle is obtuse. The angle opposite side $9$ is obtuse: $\cos C = (25+25-81)/(2\cdot5\cdot5) = -31/50$, so $C = \arccos(-0.62)$ approximately $128^\circ$.
Key Insight
In an obtuse triangle, the altitude from the obtuse angle's vertex falls outside the triangle, on the extension of the base. This is why some geometric constructions (like finding the orthocenter) require extending the sides.
Definition
An obtuse triangle has one angle $\theta > \pi/2$, equivalently $\cos\theta < 0$, equivalently $a^2 + b^2 < c^2$ for sides $a, b$ adjacent to the obtuse angle. The orthocenter lies outside the triangle in the half-plane determined by the longest side. The circumcenter also lies outside, on the opposite side of the longest side from the obtuse vertex.
Example
Triangle with sides $a=4$, $b=5$, $c=8$: $c^2=64$, $a^2+b^2=41$. Obtuse at $C$. By law of cosines: $\cos C=(16+25-64)/40=-23/40$, so $C=\arccos(-0.575)$ approximately $125.1^\circ$.
Key Insight
In the moduli space of triangles, the obtuse triangles, right triangles, and acute triangles partition the space into three regions separated by the curves $a^2+b^2=c^2$. The right triangle condition $c^2=a^2+b^2$ is a surface (curve in 2D parameter space), and obtuse triangles occupy a larger proportion of the moduli space than acute triangles.